Let's solve each of the given equations step by step.

### Equation 1: \( x - \frac{2}{x} + 1 = 0 \)

1. Multiply through by \( x \) to eliminate the fraction:
\[
x^2 - 2 + x = 0
\]
2. Rearrange the equation:
\[
x^2 + x - 2 = 0
\]
3. Factor the quadratic equation:
\[
(x + 2)(x - 1) = 0
\]
4. Solve for \( x \):
\[
x + 2 = 0 \quad \text{or} \quad x - 1 = 0
\]
\[
x = -2 \quad \text{or} \quad x = 1
\]

### Equation 2: \( x - \frac{2}{x} - 3 = 0 \)

1. Multiply through by \( x \) to eliminate the fraction:
\[
x^2 - 2 - 3x = 0
\]
2. Rearrange the equation:
\[
x^2 - 3x - 2 = 0
\]
3. Factor the quadratic equation:
\[
(x - 2)(x - 1) = 0
\]
4. Solve for \( x \):
\[
x - 2 = 0 \quad \text{or} \quad x - 1 = 0
\]
\[
x = 2 \quad \text{or} \quad x = 1
\]

### Equation 3: \( x - \frac{4}{x} + 4 = 0 \)

1. Multiply through by \( x \) to eliminate the fraction:
\[
x^2 - 4 + 4x = 0
\]
2. Rearrange the equation:
\[
x^2 + 4x - 4 = 0
\]
3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 4 \), and \( c = -4 \):
\[
x = \frac{-4 \pm \sqrt{16 + 16}}{2}
\]
\[
x = \frac{-4 \pm \sqrt{32}}{2}
\]
\[
x = \frac{-4 \pm 4\sqrt{2}}{2}
\]
\[
x = -2 \pm 2\sqrt{2}
\]
So the solutions are:
\[
x = -2 + 2\sqrt{2} \quad \text{and} \quad x = -2 - 2\sqrt{2}
\]

### Equation 4: \( x - \frac{4}{x} - 5 = 0 \)

1. Multiply through by \( x \) to eliminate the fraction:
\[
x^2 - 4 - 5x = 0
\]
2. Rearrange the equation:
\[
x^2 - 5x - 4 = 0
\]
3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -5 \), and \( c = -4 \):
\[
x = \frac{5 \pm \sqrt{25 + 16}}{2}
\]
\[
x = \frac{5 \pm \sqrt{41}}{2}
\]
So the solutions are:
\[
x = \frac{5 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{41}}{2}
\]

### Summary of Solutions:
1. \( x = -2 \) or \( x = 1 \)
2. \( x = 2 \) or \( x = 1 \)
3. \( x = -2 + 2\sqrt{2} \) or \( x = -2 - 2\sqrt{2} \)
4. \( x = \frac{5 + \sqrt{41}}{2} \) or \( x = \frac{5 - \sqrt{41}}{2} \)